Feynman Path Integral Using Operator Integration in Banach Space

Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits with several walls in terms of Lagrangian instead of Hamiltonian resulted in a magnificent work. It was known as Feynman Path Integrals in quantum physics, and a large part of the scientific community still considers them a heuristic tool that lacks a sound mathematical definition. This paper aims to refute this prejudice, by providing an extensive and self-contained description of the mathematical theory of Feynman Path Integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. About a hundred years after the beginning of modern physics, it was realized that light could in fact show behavioral characteristics of both waves and particles. In 1927, Davisson and Germer demonstrated that electrons show the same dual behavior, which was later extended to atoms and molecules. We shall follow the method of integration with some modifications to construct a generalized Lebesgue-Bochner-Stieltjes (LBS) integral of the form ( ) ,d u f µ ∫ , where u is a bilinear operator acting in the product of Banach spaces, f is a Bochner summable function, and μ is a vector-valued measure. We will demonstrate that the Feynman Path Integral is consistent and can be justified mathematically with LBS integration approach.


Introduction 1.Introduction to Young's Double Slit Experiment
In modern physics, the double-slit experiment is a demonstration that light and matter can display characteristics of both classically defined waves and particles; moreover, it displays the fundamentally probabilistic nature of quantum mechanical phenomena [1].Thomas Young in 1801 is a demonstration of the wave behavior of visible light [2].At that time, it was thought that light consisted of either waves or particles.About a hundred years later, it was realized that light could in fact show behavior characteristic of both waves and particles.In 1927, Davisson and Germer demonstrated that electrons show the same behavior, which was later extended to atoms and molecules [3] [4].
Thomas Young's experiment with light was part of classical physics long before the development of quantum mechanics and the concept of wave-particle duality.He believed it demonstrated that the wave theory of light was correct, and his experiment is sometimes referred to as Young's experiment or Young's slits.
The experiment belongs to a general class of "double path" experiments, in which a wave is split into two separate waves (the wave is typically made of many photons and better referred to as a wave front, not to be confused with the wave properties of the individual photon) that later combine into a single wave.Changes in the path lengths of both waves result in a phase shift, creating an interference pattern.
A wide-ranging interview with the legendary mathematical physicist Freeman Dyson, in which he discusses his work with Richard Feynman, his attempts to build a spaceship propelled by nuclear bombs and his controversial views on climate change.

From Double Slit to Multiple Slit and Multiple Screen Wall
In 1940, Richard P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form of double-slit experiment.It was known as Feynman Path Integrals in quantum physics [5] [6].
The scientific community considers his work a heuristic tool that lacks a sound mathematical foundation.This paper aims to refute this prejudice, by providing an extensive and selfcontained description of the mathematical theory of Feynman Path Integration.
We will use a new approach called Lebesgue-Bochner-Steiltjes or briefly LBS Integration Approach [7].

Characteristic of the Path Integral
We will review a traveling a particle-wave through double, triple, or multiple slits.
In this experiment, the light wave may be passing through one or several walls with slits from the source labeled S to a destination object called O. Feynman Path Journal of Applied Mathematics and Physics Integrals, suggested heuristically by Feynman in the 40 s, have become the basis of much contemporary physics.
Applications: 1) To non-relativistic quantum mechanics to quantum fields, gauge fields, gravitation, and cosmology.2) In areas of mathematics like topology and differential geometry, algebraic geometry, infinite-dimensional analysis and geometry, and number theory.Vectors are considered in infinite dimensional Banach Spaces.

Review of STEPS in Rieman Integration
• Input: Function f, domain, and range of the integration: there exists A R ∈ such that for any ε > 0, there exists a δ > 0 such that for any partition of the domain [a, b] into a finite number of intervals.
• Impose an arbitrary partition in the domain: Assume { }, • The approximation of objective element: • Evaluate upper and lower Riemann SUM: • Total estimated value: The corresponding Riemann SUM: is as close as we desire to a constant value.
• A constant limit exists: ( ) Or we can interpret A as the limit of the Riemann SUM when N is approaching infinity.That is, ( ) • Integral: We define the value of A when the limit in (1.2) exists, and the Riemann Integral denoted by:

Review of STEPS in Riemann Stieltjes Integral
• Input: Functions f and α, domain, and range of the integration: ∈ such that for any ε > 0, there exists a δ > 0 such that for any parti- tion of the domain [a, b] into a finite number of intervals.We accept a condition for function Y = f(x) to be a continuous function on [a, b].
• Define upper and lower Riemann-Stieltjes SUM by: Assume where ( ) ( ) Notice that when the function α is a nondecreasing function, then we can drop absolute value.
• Definition: For every, there exists a ∆-partition such that , • In this case, we use ( ) Important notice: When ( ) t t α = , then, the Reimenn-Steiltjes integral will re- duce to just Riemann integral.

Toward the LBS Integration in Banach Space
• Path Integral can be justified in a Complete Normed Linear Space (Banach Space).
• (G, * ) = Group G with a Binary Operation " * ": This set with binary opera- tion is closed, associative, with identity, and every element in this set is invertible.
• Boolean ring of binary operation on sets: Assume A and B are subsets of the power set of the abstract space X.Let us use two operations of union and intersection  and  of sets in a set V = P(X).Define:

Semiring of Subsets and Partition of an Abstract SPACE
• Lemma: Prove the symmetric differences: Semi-ring of subsets: Assume X is an abstract space and V is a family of subsets in X.For a set ( ) , , V * ⋅ is said to be a semi-ring.
• Closure under symmetric differences: Semi-ring of disjoint sets: For every set A and B in V, there exists an integer k and mutually disjoint sets

Measure (Volume) on a Semi-Ring ( ) V , , ∆ 
Assume a function from a semi-ring ( ) satisfies the following conditions: for every countable family of disjoint sets, we partition the set A into mutually disjoint sets, then, ( ) ( ) where the supremum is taken over all possible decompositions of the space.The measure (volume) is positive if it has only nonnegative values.
Norm of the positive measure: let v be a positive measure (volume) defined in a semi-ring V. Define a subspace M for all volumes :V Z µ → such that,

( ) ( )
for some constant number c and all Thus: In the following Sections 1.9, 1.10, and 1.11, we will describe Lebesgue-Bochner-Steiltjes approach to the integration in Banach Space.

Space of Simple Functions (Basic): Define a Space of Simple Function S(Y)
For all i y Y ∈ , and ( ) Vectorial Form: Assume , , , the relation (1.11) can be described by: where the characteristic function i A c can be defined by: ( ) The norm of the relation (1.12):Let , , , Notice that the symbol i y represents the absolute value of the i-th component of y.

Generalized Lebesgue-Bochner-Stieltjes (LBS) Integral
Here is a general form of presenting integration based on a bilinear operator u Journal of Applied Mathematics and Physics acting on h with a measure μ: ( ) ( ) These two operators are well defined, that is they are independent of the choice 11) where h h = ∫ .

Basic Sequence of Simple Functions and Summability
A sequence of functions and a constant M > 0 such that, The space of Summable functions: The space of summable function L(Y) is the set of all functions f which,

Some Characteristics of Path Integral
The objective is to justify the mathematics of multi-slit and multi-screen experiments in both classical and quantum senses, particularly the mathematical justification of Feynman Path Integral.
The original definition presented in (1.14) and demonstrated in Reference [7], did not present the precise definition and conditions for the integral operator.However, it was claimed that this integral operator is well defined, and it is independent of the choice of h.It was concluded that it is well-defined.
We will demonstrate that this originally undefined and not clarified conclusion has an amazing and powerful application in a variety of disciplines.
Notice that this work is just a mathematical justification according to theory of integration in Banach Space.No numerical computation or new experiment will be presented.This new approach of Feynman Path Integral is open for application to other parts of mathematical Physics for further research.Dealing with some problems in Quantum Mechanics, as we are expecting in this article, like solving the Schrodinger Equation, requires certain conditions to analyze and describe the physical phenomenon.Thus, we need some rigorous mathematical work to justify the Feynman Path Integral, which was originally accepted intuitively.Many attempts were made to bridge the gap of mathematical work; thus, much research and investigation produced some significant results.

Justification to Have Infinite Dimensional Abstract Space for Feynman Path Integral
We try to show some of the foundations and principles of the Feynman Path Integral, which requires us to begin in the following mathematical language: 1) Paths are decided to be within an infinite dimensional space.As a result, the integration will be designed in Banach Space using Bochner integration [14] [15].
3) Due to the Uncertainty Principle the trajectories must be selected from the probability distribution spaces.As a result, we may select normalized functions with orthonormal paths.
4) The Lebesgue Integration theory can be used properly for Probability distribution functions.
5) Due to many impulsive responses for a variety of operators, we need to include integration theory which can include Dirac's Delta functions.Thus, we are including Dirac's Integration System.

6)
In our investigation, we will demonstrate that Bogdan's approach toward Lebesgue-Bochner-Stieltjes integration will be a feasible approach for Feynman Path Integral.

7) Operator Integral is a necessary step toward the general definition of Using
Lebesgue-Bochner-Stieltjes integral in Banach space (see [16] and [17]).
We are looking for some integration methods in which we can cover all these characteristic conditions.In addition to the list of 1) -7), we need to apply operators to the natural phenomenon like Hamiltonian, position, momentum, …, and energy operators to be able to justify Feynman Path Integral.
In the last stage of Lebesgue-Bochner-Stieltjes integration, we try to use the following general formulation.

Integrals with Operators
Assume that Y, Z, and W are Banach Spaces.Let a be a fixed bilinear operator : u U Y Z W ∈ × → .Let us define a set of operators U such that, One example of this operator integral is the following Hamiltonian wave function: ( ) where t ∈  , and H-Hamiltonian.
For fixed operator u U ∈ , and M µ ∈ , we define a vector ,d , , , , , where j y Y ∈ , and j A V ∈ .
We would like to replace a wave-particle for the operator such that bra vector: We can define the measure on a one-dimensional x-axis: Now, let us assume that , u a b = is a vector and u is operating on two components ( ) ( ) . We may define the bilinear or trilinear operator , We can use the general operator defined in (2.2) and demonstrate the integral using the Hamiltonian operator such that, ( Example of Orthogonality of paths: Verify that two sequence functions: are orthogonal, that is 0 for , 2 for

Position Operator
We let u represent a position operator on the x-axis.As a result, when u is applied to a wave-particle ψ with Dirac's bra-kett symbolic notation: The symbol ψ * is the complex conjugate (see [18] and [19]).Eigenvalue λ and eigenvector of the position operators can be calculated: This implies that ( ) 0 x ψ = everywhere except at x λ = .This is a behavior of Dirac Delta function for eigenvector.That is, ( ) Let us apply the relation (2.3) for two different values where (i and j are integer subscripts).The integral (2.4) can be evaluated at these points: ( ) ( ) The relation (2.6) demonstrates the Orthogonality of the wave-particles.
In addition, the next relation proves the impulsive behavior, ( )d 1.

Momentum Operator
It is well known that the momentum operator acting on a wave-particle ψ can be described by: For eigenvalue and eigenfunction, we can demonstrate: This equation can be solved by separation of variables.Thus, ( ) where A is a constant of proportionality.It should be normalized, and it will be in the following form: ( ) ( ) Notice that (2.12) is a result of the product of the following using the relation (2.10) when they are acting on the wave-particle: ( )

Schrodinger Wave Equation
The Schrodinger wave equation is a fundamental equation of quantum mechanics that describes the behavior of quantum particles, such as electrons or photons.It was conceived by the Austrian physicist Erwin Schrodinger in 1925.
Derivation using wave equation: The derivation of the Schrodinger equation starts with the classical wave equation, which describes the propagation of waves.
The classical wave equation is given by, where ψ is the wave function, is the second derivative of ψ with respect to time, 2 ψ ∇ is the Laplacian operator acting on ψ, and v is the velocity of the wave.In quantum mechanics, the wave function ψ is a complex-valued function Schrodinger postulated that the wave function satisfies a similar equation, called the Schrodinger equation, but with a modified form: where H is the Hamiltonian operator, which represents the total energy of the particle, E is the energy of the particle and ψ is the wave function.
The Hamiltonian operator H is defined as: We represent the total energy of the system, where m is the mass of the particle, p is the momentum operator, and V is the potential energy, by substituting the classical wave equation into the Schrodinger equation, we obtain: To simplify this equation, we make use of the de Broglie relation, which states that the momentum of a particle is related to its wavelength by: p k =  , where 2 h = π  is the reduced Planck's constant and k is the wave number or constant vector [20] [21] [22].
Substituting p k =  into the equation and rearranging terms, we get: This is a time-independent Schrodinger equation for a particle with fixed energy E.
To obtain the time-dependent Schrodinger equation, we introduce the concept of a time evolution operator, denoted by U(t), which allows us to evolve the wave function ψ at different points in time.
Assume that a wave function is described by: The time derivative of the wave function will be:

Discussion
For a brief review of the Lebesgue-Bochner-Stieltjes (LBS) integration, we started to show the operator integration.This is a generalization of integration in Banach space, which was done by Bochner [23].Authors are not aware that any prior application of this "operator integration" to sciences or engineering education was used by others.
The operator integration (LBS) is a form ( ) , where u is a bilinear operator acting in the product of Banach spaces, f is a Bochner summable function, and μ is a vector-valued measure.
Quantum interference of different paths depending on their phases, and hence their classical action, will have all possible paths.The question is, what is the probability density of the amplitude in Δt = 10 Nano sec., of the infinite number of walls with infinite number of slits?Thus, a finite dimensional Riemann-Steiltjes integral does not work for Feyn-man Path Integral.The infinite dimensional Bochner integral which is designed in Banach or Hilbert Space is feasible for this problem.

10 )
Assume p and p' represent momentum of the wave-particles ( ) points x and x'.Using Dirac's Quantum model and identity operator: about the particle's probability density.

6 )
Journal of Applied Mathematics and Physics we will get the final form of the Schrodinger equation: time-dependent Schrodinger equation, which describes the time evolution of a particle's wave function.In summary, the Schrodinger wave equation is derived by applying the principles of quantum mechanics to the classical wave equation and introducing the concept of a Hamiltonian operator to represent the total energy of the particle.The resulting equation, called the Schrodinger equation, describes the behavior of the particle's wave function and allows for the calculation of its probabilities and energies.
1, , .7) Next, multiply both sides of the Plank-Einestein equation E ω =  by ψ → : Now, let's take the second partial derivative of the wave function (3.6) w.r.t x:The time evolution operator (3.8) satisfies equation(3.5),where i is the imaginary unit.By substituting the expression for H and rearranging terms, we get: Substituting(3.11)in this equation: